Vector calculating device



March 29, 1949. 1 RElcHE v2,465,481

` VECTOR CALCULATING DEVICE Filled Jan'` 25, 1945 3 Sheets-Sheet l 4v Zig I I Tinl. a

INVENTO @QC/e BY @W71 f5@ ATTORNEY March 29, 1949. REICHE 2,465,481

l VECTOR CALCULATNG DEVICE l Filed Jan. 23, 1945 v v3 SheetsvSheet 2 W f7 A ATTORNEY March 29, 1949. l.. REICHE 2,465,481

VECTOR CALCULATING DEVICE Filed Jan. 2s, 1945 Y v s sheets-sheet 3 INVENTOR la? w Patented Mar. 29, 1949 UNITED STATES TENT OFFICE Claims.

This invention relates to calculating devices and more particularly to apparatus for the solution Vof vector problems.

Vectors .are used in mechanics and electricity to denote forces, alternating currents and voltages and the like. The addition and subtraction of vectors for the determination of resultants is, in general, .a tedious mathematical process. A principal object of this invention is to provide simple mechanical means for eiecting rapid solutions vof vector problems.

`In elect the devices embodying the invention may becalledvector slide rules whose use enables the 4.operator to determine rapidly and accurately the solutions of vector problems involving the addition and/or subtraction of two or more coplanar vectors.

Another object of the invention is to provide devices on which data may be set up and solutions may be. ready oil directly in either the polar or the Cartesian form.

Still further objects of the invention are to provide such devices with a minimum of working parts, that may be simply made at comparat'vely low cost, that may be easily and quickly operated and that will give accurate solutions for vector problems.

To the accomplishment of the foregoing and suchv other objects as may hereinafter appear, the invention consists in the novel construction and, arrangement of parts hereinafter to be described and then sought to be deiined in the appended claims, reference beingl had to the accompanying drawings forming a part hereof which show merely for the purposes of illustrative, disclosure preferred embodiments of the device. it being expressly understood, however, that changes may be made in practice, within the scope-o1? the claims,l wthout digressing from the inventive idea.

In the drawingin which similar reference characters denote corresponding parts:

Fig. 1 is aplan view of a device embodying the invention;

Fig. im); shows amodication ofv details of the device;

Fig'. 2 is a, section taken along line 2'-2 of the device of Fig. 1 and viewed: in the direction of the arrows Fig. 3 is a section taken alongy line 3-3 of the device. of Fig. 1 and viewed in the direction of the arrows;

Fig. 4 is4 a. plan view of a slightly modified' form of device embodying the invention;

Fig, 5. .isA a. plan view oi a further. modification;

Fig. 6 is `a plan lView of ystill another modification;

Fig. 7 is a plan view of yet another modication;

Figs. 3, 9, 10 and l1 illustrate -diagrammatically the successive `positions of various of the mem bers of the device lof Figi in performing a vector addton; and

Figs. 12 and 13 illustrate diagrammatically fundamental principles of Vopera-tions regarding the calculating devices of this invention.

Referring to the drawing, and iirst to Fig f1, l0 denotes a base Vmember of suitable material for example ofgCellulodfor-'other plastic. .If desired, this base member can be opaque and p1 ovided with a mirror surface for ya purpose to be presently described. This base member l0 is provided with two or more Cartesian coordinate grid systems Ila and "I Ib veither etched, printed or otherwise applied upon its top `face, and a protractor scale I3 extendingthrough 360 similarly applied and having the intersection P of one pair of the horizontal and vertical Cartesian coordinates as its center of generation. The lines of the Cartesian coordinates may be laid out at any suitable spacing. In practice, scales of the grid systems Ha and II'I'J in the ratio of approximately 3:1 will be fou-nd necessary to perform calculations with reasonably large lengths on the base I0 and on the movable members (presently to be described) to obtain accurate results. For example, the lines of system IIb may be spaced tenths or live hundredths of an inch apart, those on scale IIa three times as far apart, and' those on the protractor` scale I3 may be dividedv into degrees and/or suitable fractions thereof.-

An arm member I4 of suitable transparent material, for example, Celluloid or other plastic, isv provided. This arm member has a pivot ii'ange I5 that extends into a circular hole: I=5a in a second arm member t6'. The second. arm' member IGT is= secured pivotally tov the. basel l0 by a plug member I'T extending into hole: Iaand. is secured to iiange I5l by an adjustable screw I'Ia so that the center ot rotation of arm IIS and that oi arm member I4: are the point P. They are rotatable independently about the said point P parallel to the top plane of the base II). Thescrew Ilmay be tightened to provide suillci'ent. friction to maintain the arms I4 and I6` in any adjusted position. Other suitablejointingv,l may be provided.

The arm I4' is provided' with a longitudinally extending slot Illa. As shown, this slot has dovetail. section tov receivea slidemernber I8 of similar material to member I4 sldably therein so that this slide member I8 may be moved longitudinally along arm I4 in said slot I4a. The member I4 may be deformed to permit insertion of slide I8 and to provide sufficient friction to maintain the slide member I8 in any adjusted position. A center line I9 is suitably applied to the under face of the slide member I 8 so that the protractor scale I 3 may be read thereby.

A pivoted member 2| of similar material as the other pivoted members I4 and I6 is pivotally secured by a suitable pivot 22 to the inner end of slide member I8 with sufficient friction to maintain any adjusted position and so that its center of rotation 23 may by movement of slide I8 be positioned in axial alignment with the center P. This center of rotation 23 is suitably applied as a reference mark. The member 2| has a longitudinally extending slot 2 Ia similar in section to slot |4a and a slide member 24 is mounted and carried in this slot'l in similar manner to that in which the other slide member I8 is carried in its arm I4. This slide member 24 has two or more linear scales 25 suitably applied to its under side and designed to cooperate with cross line 26 applied to the lower surface of member 2|. The inner end of slide member 24 also has a reference mark 21. The slide member 24 has a center line 28 along the entire length of its lower surface to be used in reading the protractor scale The pivoted arm member I6 is of material similar to member I4 and is independently rotatable about center P as described above, A longitudinally extending slot Ilia is provided in member I 6. This slot |6a has a dovetail section similar to that of slot I 4a and a slide member 29 of material similar to arm I4 is slidably carried by the slot Ilia so that said slide member 29 may be moved longitudinally along arm I6 in said slot I8a. Suiicient friction is provided to maintain the slide in any adjusted position. The slot I6a at its inner end terminates short of the hole I5a. Linear scales 30 are suitably applied to the under surface of slide member 29. Cross line 3| is applied to the lower surface of arm I6 adjacent its outer end so that the slide member 29 may be adjusted to any desired position with respect to the cross line 3|. The slide 29 likewise has a center line 32 along the entire length of its lower surface so that the protractor scale I3 can be readftherewith. It likewise has a reference mark 32a at its inner end.

`The arm members I4, I6 and 2| may have rounded edges to prevent sticking when moved relative to each other.

,The three slide members I8, 24 and 29 are provided at their outer ends with suitable manipulating knobs 33, 34 and 35.

structurally the modication of Fig. 4 is identical with that of Figs. 1 to 3 and identical reference characters denote corresponding parts. The additional feature in this structure is the provision on the under face of slide I8 of linear scales'36. Likewise the cross line 38 is applied to the lower surface of member I 4 on which scales 3B maybe read.

Likewise, structurally, the modification of Fig. 5 is identical with that of Figs. 1 and 4 and identical reference characters denote corresponding parts. The difference in this modiiication over that of Fig. 4 is the elimination of the scales 3|), as well as the cross line 3 I.

The modiiication of Fig. 6 resembles that of Fig. 5 except that the member I6 and slide mem- LII ber 29 are eliminated. Corresponding reference characters denote identical parts. In place of the eliminated members I6 and 29, an arm member 40 is pivotally secured to the under face of the base Ill so that its center of rotation is the point P. This arm member 40 extends outwardly beyond the periphery of the base member IIJ. Another arm member 4| is pivotally secured at 42 to the arm member 40 so as to be movable over the top face of base member I0. This arm 4| has a suitable reference mark 43 applied to its under face for a purpose to be presently described. Arm members 4I] and 4| are secured at their joining points with friction joints so that they Will retain any adjusted position. Likewise these members are preferably of transparent material similar to that of the other members I4, 2 I, I8 and 24,

The further modification of Fig. 7 resembles that of Fig. 6 except that the arms 4U and 4| are eliminated. Corresponding reference characters denote identical parts. In place of arms 4U and 4I, the peripherally located flange Ib is provided on the base I0. A link member 45 is pivotally secured at 46 to the ange IIlb. Additional link members 4`I and 48 are provided, link member 4l being pivotally secured at 49 to link member 45, and link member 48 being pivotally secured at 59 to the link member 41. The link members 45, 4'I and 48 are preferably of transparent material similar to that of the other members I4, 2|, I8 and 24. A suitable reference mark 5I is applied to the under face of link member 48 for a purpose presently to be described. The pivot joints 46, 49 and 50 are made suiliciently tight frictionally to maintain the links 45, 41 and 48 in any adjusted position.

The modifications of Figs. 6 and 7 are advantageous in that they have greater mechanical stability, less parallax effects and are simpler to construct.

The devices of Figs. 4-7 inclusive all may be used in identical ways to transform a given vector from polar coordinate form to Cartesian form and vice versa. For such operations only the base I0, arm member I4 and slide I9 of each thereof need be used.

For example, given the vector A a of Fig. 8, to obtain the corresponding Cartesian coordinates of such a vector, the vector Aa is set up by rst rotating the arm I4 about center P until the center line I9 indicates the angle a on protractor scale I3. Thereafter the slide I8 is pulled out in its slot I4a until the cross line 38 on member I4 indicates on the scales 36, the magnitude lAI. This positions the reference 23 with respect to the base IIJ at a position corresponding to III-I-iy. The values of a and y are then read directly on the scale I Ia or IIb giving the solution of the transformation of polar form A a to Cartesian form :c-I-iy.

Conversely, ii the Cartesian form -I-jy is given, the polar form Aa is readily formed by manipulating the arm I4 about its pivot and the slide member I8 to position the reference 23 at a point on scale IIa or IIb of base ID corresponding to -I-i'y. The polar transformation is then read directly, IAI being the Value on scales 36 under cross line 38 and a being the value on protractor scale I3 under center line I9.

The same results may be achieved in the device of Fig. 1 by manipulating the arm IBand slide member 29 in a manner corresponding to that described for arm I4 and slide member I8 of Figs. 4-'7 inclusive. This arm I3 must be used inFig. =1 insteadof arm I4 becausethere are no scalesonthe slide |8of Fig. 1.

'Toperform a 'Vector addition orrsubtraction-o'f two'fvectors Abr and B with the devices -of Figs.4-7 inclusive, with reference-,to Fig. 12,.the slide member I8 of these figures is pushed in slot Mato-position reference 23 over the center P, and (I) arr'n 'lll is rotated about center P until the angle -a is read onprotractor scale Iiunder centerline i9. (II) Then arm 2l isrotatedabout P-as-acenter until ythe angle is read on protractor scale IS-undercenter-line28. (III) Maintaining this angular position eiarm 2l,.the-.slide 24 ispulled-'outfuntil.the-magnitude [Bl is read on scales 2li `unde: cross 4line 25. (IV) vThen still maintaining thefangular position of arm 2l, the slide i8 'or arm lill is pulled out along slot Mia untilthe rnagnitude|A| appears on scales 'under cross line 38.- lThis positions the reference 21 of slide 24 so that a Cartesian coordinate solution of A or--B may be read directly on the scales llila; or l lb of base l'. Thus the .rand g/ coordinates of reference 21 with respect to scales Maand llb give thefnurnerical'values :r and 'y' of the'sol'ution in Cartesian coordinates and the resultant R' may be written oi as R=-..ri,7y. In lig. 4 the-polar solution is then easily determined by-means of member l.

Inorder tond'thePolar coordinate solution in Figs. `fato" 7, it is'necessary to 'free the arm Ul foi-further motion. In the'case of Fig. the arm i8 andfslide member 29 are manipulatedto bring the reference 32a thereoiF into alignment with the reference v2. (V) Then the l arm lil is rotated and itsslide member i3 manipulated to bring lits reference a intoalignment with positioned reference 32a. The lpolar value of R is new read directly 'fromscales 3B under-cross line Bi'of arm I-ll'and the angular Value 1;/ of lthesolution R u `is read on protractor scale i3 below vcenterlline i9. The arm l5 serves simply the function-of a keeper i. e. to free the arm lll for further manipulation while keeping the position of reference .2l determined by addition of vector Awl-B In the cases of use-ofthe devicesofFigs. 6 and 7 which do notghaveany arms fthe arms di? and lll l.of Fig. 6 or links (l5, el t vfunction as keepers afterthe position o reference 2l -determined by the `addition of vectors Aa{-B has been located. In the case of Fig. v(5, di) and di'. are manipulated to bring reference linto alignment with reference il?, and similarly in Fig. 7, links dl and i3 are manipulated to bringyreference 5l into alignment withy reference 2l?. lThis frees the arms Irl lof each of these devices -so that they can be adjusted respectively tro-bring references 23 into alignment either with ythe adjusted references or 5l. The values of R' and d can then-be i'eaddirectly,r respectively on scales .iliunder cross line 55E of arni le and protractor scale i3 under center line I9.

Subtractionof Vectors A01-,BUS issirnilariy performed since in effect The operational `steps for substraction are shown diagrammatieally in Fig. 13. (I) rlShe Islidenr-iemb'erlll ofthese` Figures 2 7 inclusive is lpushed in slot 'Ilia to the positionreference 23 vover the center Pand arm It is rotated about centerPfuntil-the;anglea is read on protractor scale; lf3: under'center line l. (II) Thenarrn?. l is ;rot ated about 1P vas a V,center until ithe angle v)Het186.isfread.:curprotractomscale Iunder .center line 28. f (III). Maintaining. this @angular pos-i.

tionfof arml l, thelslideLZis pulled-out Auntil-'true magnitude IBlis 'readon scales 25'underxcross line E. I(IV) Thenstill-maintaining-'the angularposition orarms M and2l, the slide ISis pulled-out alongslot Usa until the magnitude 1A| appears on scales under the cross line 3.8. This positions the reference ,2 .of 'slide LN so that a Cartesian coordinatesolution of. 'A @E 3B may be lreaddirectly on the scaleslla or' l lb of base l0. Thusthe :c and coordinatesl ofreference 'f'iwvithrespect toscales Ha vor lib give-the numerical value :13" and y" of the solution in Cartesianscoordinates and the resultantrmay be Written oir as .'R=w'1iy" x-as the case .may be. In Fig. 4 the polar solutionisthengeasilydetermined.

'(fV) jn lorder,v to find. the' Polar-'coordinate solution in Figs. Llxto 7, it is necessary to reezthe arms lf3 for further motion exactly asin the case of` addition. `This is .done by 1 manipulating the keeper-:arm It. of Figli` inthe/same way asior a'ddition, .an.d similarly t-hefr keepers Aditi of @and 135,151, a8 of Fig. r{,wlience the values 4of R" and 0.can be read directly, respective-ly on scales undercrosslinesjS of the -a-rm lll-and on'protractor or polar `scalevll under center line i9 after the arms hayefbeen adjusted inthe .saine Waves ip't-be'case of vector addition.

elf .additional :vectors vare to be added or sub" trafoted, aforexample 4third .vector is to be added tor subtracted from the addition ,orjsubtraction ;of 'Aa.and1B-, the third vector is added or subtracted from Athe-solutions'R` up or R":0 .previouslyobtained in substantially the same :manner as described for the other additionsorsubtractions.

Thedevice of Fig. `ljrnay -be operated, forexample, itogsecure-the .vector sum of two vectors Abr and '.Bff as vshovvngin Figs.V 8-11 inthe following` manner:

4Arm member 1 6 is rotated about-center P until its-centerline32 indlcatesgon polar scale I3 the angle la. {II-hen slide 29 is moved outwardly in i slot lapuntil its scales V3! -read through cross liner-3| indicates the'value [Al of the vector A a. These operations position the reference 32a with respect .tothe base l0 at .a point ycorres ,ponding tosthatwwhich the leading tip ,of the vector A01 would-occupy: (-Fig) The center referenQc-ZS :of arm member `tl is then moved into axial alignment With the center of rotation P and arm member I4 is rotated about center lP ato overlie Athe adjusted position of lthe arm member I5. 'Then arm memberZll isrotateld about v.the axially `aligned centers r(P land l2.3,) until Athecenter line 28 of its slide2 4 indicates on the protractor scale .[3 the-angle Then slide member Mis pulled :outwardly of slot-,2da until the,cross.linezindicates on scale .f2.5 the value:]B| ofthe vector B 6. These `operations position the reference markl'! on Aslide 2i=with respect'ztofthe base :lll atxa pointcorresponding to f that `which the leading tip of` the y,ector1;B, would occupy (Fig.\9),.

Thenwith the three armmembers M, lB-gand 2| maintained in :their angularly 4adjusted posi" tions :as just described, fthe :slideymember `lill of arm .member ,Ul ris pulled `outwardly Nin `its -slot Ma .untilrthe referelmev mark 12,;3 ovclles reference fmark @3.211, AThis .positions the arm :2l :as shown :in Fig. 10, ,The `4reference mark .Trigon lies-.attire position of the-leading end` ft 13esultant of sum of the two Vectors.Agagandg in their adjusted positions, arm member I6 is rotated about point P and slide member 29 pulled out in its slot Ia until the reference 32a underlies the position of reference 21 as shown in Fig. 10. The value of the resultant R is then read directly on scales 30 of slide member 29 under cross line 3l and the angular disposition .'1 of the resultant is read on protractor scale I3 under center line 32 of slide 29. In other Words the distance P to 32a of Fig. 11 is the numerical value of the resultant at the resultant angle. In this way Aa+B=R 4f can be conveniently determined. If additional vectors are to be added for example A a|B-|Cp, vectors A and B are added as before and their resultant Rb/f is added to Cp in the same manner.

To determine the diierence between two vectors, for example, Aa-B with the apparatus described in Fig. l would be an equally simple series of operations.

The arm member I5 is positioned as shown in Fig. 8 to bring the reference 32a to the position of Aa. Then arm I4 is rotated into parallelism with arm I6 and slide I 8 is positioned to bring reference 23 over center P and arm member 2| is rotated to bring it to the position ,3+ 180 and slide 24 adjusted so that reference 21 is positioned at the value |B|. Then sl-de member I8 is adjusted to bring reference 23 into alignment with reference mark 32a. Then arm member I6 is rotated and the slide member 28 is manipulated to move reference 32a into alignment with reference 21. The scale 30 under cross line 3| then indicates the numerical value of the difference between the two vectors and the reading under center line 32 on the protractor scale I3 indicates the angular disposition of the resultant vector. The real and imaginary components of this vector may be read directly on scale II.

It will be noted that on the devices of Figs. 5-1 the keeper arm keeps the sum Abri-Bg?, whereas in the device of Fig. l arm I4 keeps vector A a only. That is the only difference.

The devices of Figs. 1-7, therefore, aords very convenient means for adding and subtracting coplanar vectors. In a sense they afford quasigraphical addition and subtraction of vectors. rlhey will also handle problems in plane trigonometry (including certain algebraic problems that may be reduced to trigonometric problems, such as the law of Pythagoras:

)\=\/a2-IbE It should be noted that in the above descriptions the term vector was merely used as a general mathematical concept, not to conne the possible applications of the devices described to vector calculations alone.

The various center lines I 9, 28, 32, scales 36, 25, 30, cross lines 38, 26 and 3l, and reference marks 23, 21, 32a, 43 and 5I are positioned on such faces of the members on which they are applied so that parallax errors in reading of scales and references will be as small as possible. Likewise if the base I is opaque, the mirrored surface thereof will eliminate parallax when aligning two references and in reading various of the scales.

The reference marks 23 and 21 are, preferably, cross lines that intersect each other at angles of 70 while reference marks 32a, 43 and 5I are a y with legs at 120 so as to make it easy to read the scales on the grid system and to superpose the reference marks with respect to each other when necessary.

It will be understood that the particular arrangement of the scales and reference marks can be varied on the various arm members and slides. For example, as shown in Fig. 1a, the scales 30' whichare equivalent to the scales 30 on slide member 29 are provided instead on the arm I6'. The y reference 32a' is augmented by additional indicators, as shown, so that the reference 3| on arm I6 of Fig. 1 can be eliminated. This same system can be utilized for the other slide members With respect to the arms which carry them if desired. Other arrangements of the scales and indicators are possible and contemplated as within the scope of this invention.

The devices described have many other practical uses. While specific embodiments have been described, it should be understood that changes in structural detail are contemplated within the scope of the claims. There is no intention of limitation to the exact details shown and described.

What is claimed is:

1. A calculating device of the character described comprising a base member having a polar scale and rectangular grid systems thereon, transparent slotted members pivotally supported on said base member to rotate about the center of said polar scale, transparent slide members carried by the pivoted members and movable longitudinally in the slots thereof, an additional transparent slotted member pivotally carriedby one of said slide members whose center rotation is movable into axial alignment with said firstnamed center and a longitudinally movable transparent slide member carried by said additional member in the slot thereof and scales and indicators in conjunction with various of said pivoted and slide members.

2. A calculating device of the character described comprising a base member having a polar scale and rectangular grid systems thereon, an arm member pivotally supported on said base member to rotate about the center of said polar scale, said arm member being ixed against longftudinal movement relative to said center, a slide member carried by the pivoted arm member and movable longitudinally thereof, a member pivotally carried by said slide member and whose center of rotation is movable into axial alignment with said first-named center, a longitudinally movable slide member carried by said second pivoted member, an additional pivoted member adapted to rotate about said rst center, and a slide member movable longitudinally with respect to said additional pivoted member and having a reference, said two last-named members serving as a keeper for xing a point determined by others of said pivoted and slide members.

3. A calculating device of the character described comprising a base member having polar and rectangular scales thereon, two transparent arm members both pivotally supported on said base member on the same face thereof, to rotate about the center of the polar scale, said arm members being fixed against longitudinal movement relative to said center, transparent slide members carried by the pivoted members and movable longitudinally thereof, scales and indicators in conjunction with all said transparent members, an additional member pivotally carried by one of said slide members whose center of rotation is movable into axial alignment with said first-named center, a longitudinally movable transparent slide member carried by said additional member and scales and indicators in conjunction with said additional member and said last-named slide member whereby said polar and rectangular scales may be observed through the various members.

4. A calculating device of the character described comprising a base member having polar scale and rectangular grid systems thereon, an arm member pivotally supported by said base member to rotate over its surface about the center of said polar scale, a slide member carried i by said pivotally supported arm member, said slide member having a xed reference thereon for cooperation with said rectangular grid system and a fixed reference thereon for cooperation with said polar scale and one of said members having a linear scale whereby said two members may be positioned readily to place said rstnamed fixed reference with respect to said center at a point corresponding to the numerical value of and angular direction of a vector, a second pivoted arm member supported by said base at said center, a slide member carried by said second arm member, said second slide member also having a Xed reference thereon for cooperation with said rectangular grid system and also having a xed reference thereon for cooperation with said polar scale, and one of said two lastnarned members having a linear scale, a third pivoted arm member supported rotatably about the first xed reference on said first slide member, its center oi rotation being movable into axial alignment with said rst-named center, a third slide member carried by said third arm member, said third slide member having a iiXed reference for cooperation with said polar scale, and one of said two last-named members having a linear scale whereby said third member may be positioned in the angular direction of a second vector and then have its center of rotation transposed to overlie the position of the rst xed reference of said iirst slide member when the first fixed reference lies at the said point determined by the value and angular direction of said first-named vector, said first-named arm member then being movable to the angular position of the resultant of said two vectors, and its slide member then being movable to bring the iirst xed reference thereof to overlie the point corresponding to the numerical value of said resultant at the angular disposition of the latter whereby the numerical value of said resultant may be read directly on the linear scale associated with said first members, and its angular value by the position of the second fixed reference of Said rst pivoted member with respect to said polar scale, and its real and imaginary components may be read on said rectangular grid system.

5. A calculating device of the character described comprising a base member having a polar scale and rectangular grid system thereon, an arm member pivotally supported by said base member to rotate over its surface about the center of said polar scale, a slide member carried by said pivotally supported arm member, said slide member having a fixed reference thereon for cooperation with said rectangular grid system, and having a second iixed reference thereon for cooperation with said polar scale and one of said members having a linear scale whereby said two members may be positioned readily to place said first-named fixed reference with respect to said center at a point corresponding to the numerical value of and angular direction of a vector, a second pivoted arm member supported by said base at said center, a slide member carried by said second arm member, said second slide member also having a iixed reference thereon for cooperation with said rectangular grid system and having a second lixed reference thereon for cooperation with said polar scale, and one oi said two last-named members having a linear scale, a third pivoted arm member supported rotatably about the rst reference on said rst slide member, its center of rotation being movable into axial alignment with said rstnamed center, a third slide member carried by said third arm member, said third slide member having a iXed reference for cooperation with said polar scale, and one of said two last-named memr. bers having a linear scale whereby said third member may be positioned in the angular direction of a second vector and then have its center of rotation transposed to overlie the position of the rst fixed reference of said first slide member when the iirst fixed reference lies at the said point determined by the value and angular direction of said rst-narned vector, said first-named arm member then being movable to the angular position oi the resultant of said two vectors, and

"1 its slide member then being movable to bring the iirst iiXed reference thereof to overlie the point corresponding to the numerical value of said resultant at the angular disposition of the latter whereby the numerical value of said resultant may be read directly on the linear scale associated with said first members, and its angular value by the position of the second Xed reference of said first pivoted member with respect to said polar scale, and its real and imaginary components may be read on said rectangular grid system, and frictional means for maintaining each member in any adjusted position.

LUDWIG REICHE.

REFERENCES CITED The following references are of record in the le of this patent:

UNITED STATES PATENTS Number Name Date 1,296,704 Sayre Mar. 11, 1919 1,560,747 Weinbach Nov. 10, 1925 1,853,740 Mueller Apr. 12, 1932 1,881,204 Lundgren Oct. 4, 1932 2,244,945 Goodale June 10, 1941 FOREIGN PATENTS- Number Country Date 744,709 France Apr. 26, 1933 

